University of DerbySchool of Computing and Engineering |
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In-course Assignment Specification |
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Module Code and Title: 6CC504 – Cryptography & Coding |
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Assignment No. and Title: Portfolio Part 1. Cryptography |
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Assessment Tutor: Lee Barnby |
Weighting Towards Module Grade: 50% |
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Date Set: Thursday 3 March 2022 |
Hand-In Deadline Date: Monday 25 April 2022 (23.59) |
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Penalty for Late Submission Recognising that deadlines are an integral part of professional workplace practice, the University expects students to meet all agreed deadlines for submission of assessments. However, the University acknowledges that there may be circumstances which prevent students from meeting deadlines. There are now three distinct processes in place to deal with differing student circumstances: Assessed Extended Deadline (AED) Students with disabilities or long-term health issues are entitled to a Support Plan. Exceptional Extenuating Circumstances (EEC) The EEC policy applies to situations where serious, unforeseen circumstances prevent the student from completing the assignment on time or to the normal standard. www.derby.ac.uk/eec Late submission Requests for late submission will be made to the relevant Subject Manager/Module Leader in their School (or Head of Joint Honours for joint honours students) who can authorise an extension of up to a maximum of one week. www.derby.ac.uk/eec |
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Level of Collaboration: Individual |
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Learning Outcomes covered in this Assignment: Evaluate the appropriateness of approaches to cryptography Understand the mathematical foundations of public key cryptography Use a public key system |
Part 1: Answers to questions 1, 2 and 3 (50%)
Marking Criteria
To obtain above 70 University % credit points:
All questions will have been attempted in full, an excellent understanding will have been shown, with good solutions or examples.
To obtain between 60 and 69 University % credit points:
All questions will have been attempted, with few missing parts, a good understanding will have been shown, with reasonable solutions or examples.
To obtain between 50 and 59 University % credit points:
There will have be a reasonable attempt at answers to most of the questions, some gaps in understanding may be evident and full solutions could be missing in part and/or examples might not demonstrate the points intended.
To obtain between 40 and 49 University % credit points:
Some attempt is made at all the questions with at least some relevant understanding displayed, solutions and/or examples are missing.
To obtain between 35 and 39 University % credit points:
Work shows some relevant understanding and/or attempt at providing solutions or examples but falls short of the above.
To hand in at the first submission point.
Exercise 1 – Historical Ciphers
Development of historical ciphers, beyond the simple Caesar cipher, aimed to improve their security. Explain the ways in which knowledge of the type of content in the message can reveal weaknesses which allow the key to be discovered and provide an example.
(6 Marks)
The following message has been intercepted:
“q vquv gu q bgcx fa ugycqhtsz afs q uhszqn fa xqhq hvqh
szdszuzchu hvz wfchzchu. hvz wrfuzuh szqr-rgaz qcqrfytz
mz wqc hvgcb fa gu q hqndzs-zkgxzch uzqr fc q ufahmqsz
dqwbqyz ga jft fdzc hvz pfi (wvqcyz hvz agrz), gh gu
xzhzwhzx. q wsjdhfysqdvgw vquv atcwhgfc gu q dsfwzxtsz
hvqh hqbzu qc qspghsqsj prfwb fa xqhq qcx szhtscu q agizx-
uglz pgh uhsgcy, utwv hvqh qc qwwgxzchqr fs gchzchgfcqr
wvqcyz hf hvz xqhq mgrr wvqcyz hvz vquv kqrtz. hvz gxzqr
wsjdhfysqdvgw vquv atcwhgfc vqu afts nqgc dsfdzshgzu: gh
gu zquj hf wfndthz hvz vquv kqrtz afs qcj ygkzc nzuuqyz,
gh gu xgaagwtrh hf agcx q nzuuqyz hvqh vqu q ygkzc vquv, gh gu xgaagwtrh hf nfxgaj q nzuuqyz mghvfth wvqcygcy ghu vquv, qcx gh gu xgaagwtrh hf agcx hkf xgaazszch nzuuqyzu mghv hvz uqnz vquv.”
Find a table showing the frequency of usage of letters in the English language. Use this, and other similar knowledge of English, to devise an approach to decrypting the message, setting out the proposed algorithm.
(7 Marks)
Use your approach to decrypt the message showing your working and, in particular, noting where you had to depart from or modify your original algorithm. In all cases it must be completely clear how you obtained your final answer.
(7 Marks)
Hints: the final decrypted text may not be perfect.
You are not asked to produce a working general program that can deal with all cases of this type of problem. Your approach may combine choices made ‘by hand’ with substitutions made by a program.
Total marks: 20
Exercise 2 – RSA Algorithm
The example below illustrates the operation of a public key cryptography scheme:
Suppose that Bob chooses n = 1943 = 29 x 67. Then (n)= 1848, and if he chooses e = 701, then d = 29 is a solution of:
1 = e x d – 11 x 1848.
Therefore, (n,e) = (1943, 701) is his public key and d = 29 is his private key. Now suppose that Alice wants to send the message power to Bob. She must first convert this to numerical equivalents. She chooses Table 1 below, so we are using base 26 integers.
Since n is small here, she decides to use message blocks of 2-digit, based 26 integers (with zeros packed to the right in the last block if necessary), and encipher each block separately.
First, we get the base 26 equivalents for the plaintext as 15, 14, 22, 4, 17 then we break this up as follows:
m_{1} represents numerical value for po so m_{1} = 15 x 26 + 14 = 404;
m_{2} represents numerical value for we so m_{2 }= 22 x 26 + 4 = 576;
m_{3} represents numerical value for ra so m_{3 }= 17 x 26 + 0 = 442;
with the a=0 packed to the right in the last block.
Thus Alice enciphers: 404^{701} ≡ 1419 (mod 1943), 576^{701} ≡ 344 (mod 1943), 442^{701} ≡ 210 (mod 1943)
Bob recovers the plaintext via his private key:
1419^{29} ≡ 404 (mod 1943);
344^{29} ≡ 576 (mod 1943);
210^{29} ≡ 442 (mod 1943).
He then rewrites each deciphered block as 2-digit base 26 integers and recovers the English plaintext via Table 1.
Table. 1.
[based on Ex. 3.10 “RSA and Public-Key Cryptography”, R. A. Mollin p 63]
————————End of Example, Questions Follow——————-
a) Assuming the same values for n, d, e and using the same table for the numerical equivalents:
i) Find the encrypted message when the word good is the message.
(3 Marks)
ii) Find the original message sent when the encrypted message received is 1437 149
(4 Marks)
b) Repeat the example with a different public and private key pair of your choice:
i) Show encryption of a short (6-letter) phrase.
(6 Marks)
ii) Comment on whether your choice would be more or less secure than the key in the example.
(2 Marks)
Your solution to parts a) i-ii and b) i should show how you obtained the answers, either by setting out mathematically the iterative steps for the modulus operation or by using a program which you have written to do this.
Total marks: 15
Exercise 3 – RSA Algorithm weaknesses
In the real-world implementation of the RSA protocol the keys are involve much larger numbers, usually expressed by the key length in bits (e.g. 2048), and security relies on the difficulty of discovering the prime factors of these large numbers. Nevertheless, attacks on this encryption scheme have been discovered or developed based on reducing the difficulty from the apparent one.
Find out, and write about one of the following:
The importance of good random number generation for key security.
Weaknesses introduced by the choice of prime numbers used in forming the public key.
Any other mathematical weakness of your choice.
You should refer to any exploits or attacks which use the area you have chosen to write about. In addition, you should demonstrate your understanding by incorporating a (simplified) worked example. Please use good citation practices and refer to your sources, avoiding the problem of plagiarism.
Guideline: 400-600 words
Total marks: 15
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